Method for reconstructing linear structures present in raster form

ABSTRACT

Linear structures are used to identify persons. In order to be able to combine a multiplicity of such linear structures in a database, their original images are analyzed and reconstructed using orthonormal basic functions. A preferred direction of the linear structure is determined for each pixel. A quality measure is used to evaluate the reliability of the analyzed data. Singularities (SI) and minutiae (MI) are extracted and stored.

BACKGROUND OF THE INVENTION

The invention relates to a method for reconstructing linear structurespresent in raster form. Such linear structures can be used to identifypersons. In order to identify persons, use is usually made of theirfingerprints, but also of other linear structures such as those of theretina, the vascular plexus of the retina of the eye. Use can also bemade of the linear structure in the iris of the human eye for thepurpose of a unique identification of a person.

In order to be able to use the said linear structures to identifypersons, a multiplicity of such linear structures must be combined in adatabase. A linear structure of a person to be identified is thencompared with the content of this database. In the meantime, ever morevoluminous collections of linear structures are exceeding thepossibilities for carrying out visual comparisons in an acceptable time.However, because of the rapid developments in the fields of storagemedia and the techniques of digital image processing, tools areavailable which permit the design of powerful recognition systems.

Barry Blain, Introduction to Fingerprint Automation, Home Office PoliceDepartment, Police Systems Research & Development Group, Publication No.1/93, 1993 discloses a method by means of which the line directions canbe determined. In order to determine the line direction, the image isparqueted into slightly overlapping subregions with a size of 20×20pixels, four to six pixels overlapping at the edge of the subregions.After determination of the direction, which is performed by regionalFourier transformations, a directional smoothing with subsequentbinarization is undertaken. The threshold values for the binarizationare determined by histograms in the subregions. The binary image therebyproduced still contains prominent defects, which are later corrected bymeans of diverse consistency criteria. It is not always possible toexpect an error-free correction of the defects, for example the mergingof regions which do not belong together, or the separation of lines.

In order to achieve a high processing rate in the recognition of linearstructures, the storage of essential information such as, for example,the precise position and shape of lines as well as their thickness, isfrequently dispensed with during the storage of linear-structure data.Moreover, the linear structures to be coded are frequently present onlyin a reduced image quality. In subregions, the image is for exampleblurred or has insufficient contrast. In the case of coding inaccordance with the prior art, the linear structure in such regions isestimated. However, it is then no longer possible to recognise from thestored binary image of the linear structure the degree of reliabilitywith which the individual image sections were generated from the basicimage pattern.

SUMMARY OF THE INVENTION

It is the object of the present invention to specify a method forreconstructing linear structures present in raster form, in which areliable linear-structure coding which retains all the essentialinformation of a linear structure is achieved in conjunction with aslittle computational outlay as possible.

By contrast with known automatic recognition systems for linearstructures, in the method according to the invention essentialinformation such as, for example, the precise position and shape oflines as well as their thickness is not dispensed with. By applying themethod steps, the real linear structure is transformed into an idealmodel with a locally parallel, one-dimensional linear structure, theideal model being applied separately for each pixel. Conclusions can bedrawn on the reliability of the ideal model at each pixel with the aidof the vector coefficients, which are determined as a responsecharacteristic. The vector coefficients permit compact storability ofthe linear structure.

In accordance with a development and refinement of the method accordingto the invention, smoothing of the vector coefficients determined iscarried out. A smoothed variation in the preferred direction of lines isthereby achieved.

In accordance with a development and refinement of the method accordingto the invention, the method is repeated by means of at least a secondbinomial function A(x,s) which is assigned to a different magnitudescale. Various reconstructions of the original linear structure are thenpresent.

In accordance with a development and refinement of the method accordingto the invention, the respectively best reconstruction result for eachpixel is selected by means of generating a weighting function, therespective selection result being stored in a memory as a grey-scaleimage and as a directional distribution. The best reconstruction isthereby selected from the various reconstructions of the original linearstructure for each pixel. The width of the lines and their mutualspacing in the ideal model can be varied thereby. There is thus arefinement of the ideal model.

In accordance with a development and refinement of the method accordingto the invention, the absolute value of each vector coefficient isformed and is stored in the form of a quality map. A measure of qualityis thus available for each pixel. This measure of quality can, on theone hand, be used later to establish the reliability of the linearstructure. On the other hand, characteristic features of the fingerprintcan be determined with the aid of the quality map. Smoothing of themeasure of quality, produces coherent regions of good quality, such ascorresponds to the sensitivity of the human eye.

In accordance with a further development and refinement of the methodaccording to the invention, segmentation of the reconstruction result isundertaken by means of threshold-value treatment of the quality map, inwhich all the pixels having a quality which is higher than a thresholdvalue are assigned to a valid linear structure. The existence of thelinear/valley structure in an image region can thereby be recognized ina simple way.

In accordance with a further development and refinement of the methodaccording to the invention, singularities are determined from thedirectional field with the aid of a method based on the theorem onwinding numbers. Checking of the existing directional information in theenvironment of a singularity is performed, in order to check theauthenticity of the singularity. The authentic singularities can thus berecognized and stored in a simple way.

In accordance with a further development and refinement of the methodaccording to the invention, a binary image is generated from thegrey-scale image, by means of a reconstruction of all the pixels, bysetting the limiting value for distinguishing between white and black atthe grey-scale value of 0. By comparison with the grey-scale image, sucha binary image requires little storage space and can be outlined, withthe result that the line width is reduced in steps to the width of apixel. In a sequential selection of pixel image sections, of size 3×3pixels each, minutiae can be recognized from the binary image outlinedin this way. During comparison of the pixel image sections, thesebranchings or line ends are recognized by means of pixel patterns of thesame size which contain the structures under search. The associatedpixel coordinates and the type of the minutiae are stored for later usein the event of a positive result of comparison. The reliability of thefound minutiae can be determined with the aid of an evaluation of theline quality in the close environment of the minutiae.

In accordance with a further development and refinement of the methodaccording to the invention, at the start of the method, contrastenhancement of the image information present in raster form isundertaken by means of psuedologarithmic transformation. Optimumconditions are thus created for the further method steps.

It is thus possible even for fingerprint data to be filtered withspatially variant, one-dimensional directional filters by means of themethod according to the invention, that is to say by the use of theideal model. These filters have a close relationship to Gabor filters.The spatial variance with respect to direction and dimension is achievedby linear superposition of a number of spatially invariant filters. Thesame filters are used in order to determine the smoothed directionalfield and the relevant region of the total image. The responsecharacteristic relating to the basic function is also used for thepurpose of calculating a measure of quality for each pixel. This conceptis possible on the basis of the controllability and the orthonormalityof the elementary basic functions.

BRIEF DESCRIPTION OF THE DRAWINGS

The features of the present invention which are believed to be novel,are set forth with particularity in the appended claims. The invention,together with further objects and advantages, may best be understood byreference to the following description taken in conjunction with theaccompanying drawings, in the several Figures of which like referencenumerals identify like elements, and in which:

FIG. 1 shows a block diagram of a fingerprint recognition system,

FIG. 2 shows a block diagram of the method sequence,

FIG. 3 shows a set of Cartesian basic functions,

FIG. 4 shows a set of wavetrain basic functions,

FIG. 5 shows an original fingerprint image,

FIG. 6 shows a fingerprint image preprocessed by contrast enhancementand global grey-scale correction,

FIG. 7 shows a grey-scale representation of a distribution of the imagequality,

FIG. 8 shows a grey-scale representation of the result of thesegmentation of the fingerprint image,

FIG. 9 shows a grey-scale representation of the field of the sine andcosine components of directional vectors,

FIG. 10 shows a grey-scale representation of the field of the arc ofdirection vectors,

FIG. 11 shows a grey-scale representation of singularities in thefingerprint image,

FIG. 12 shows a grey-scale representation of the filtered fingerprintimage,

FIG. 13 shows a binary representation of the filtered fingerprint image,

FIG. 14 shows a binary representation of the outlined fingerprint image,and

FIG. 15 shows a binary representation of a result of coding, includingglobal singularities.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

The block diagram in accordance with FIG. 1 shows the essentialcomponents of an automatic fingerprint recognition system. An image IMof a fingerprint, which has been read-in by a scanner and converted intoa rasterized, two-dimensional item of grey-scale information istransferred to a coding device EC. In a way to be described later, theimage IM is conditioned there into the form of data which can be usedfor automatic fingerprint recognition. If these data are to be availablefor later comparison processes, they are stored in a memory MEM whichserves as database. If, however, the data are to be used only for onecomparison process, it is sufficient for them to be stored in a memoryMEM1 assigned to the coding device EC. If required, the data can also bestored in the two memories MEM, MEM1. The data contained in the memoriesMEM, MEM1 can be compared with one another with the aid of a comparatorMT, which is coupled to the memories MEM, MEM1.

As a first step, the aim now is to explain the considerations on whichthe coding is based by means of a local image model.

The One-dimensional Continuous Theory

The core of the theory is the approximation by means of which thetwo-dimensional grey-scale image information is simulated locally. Thisis performed by means of a set of basic functions which fulfil therequired conditions in an optimum way and in so doing are calculable toa high degree. It has been shown under very general preconditions in"Generic Neighbourhood Operators", J. J. Koenderink, A. J. Doorn, IEEETransactions on Pattern Analysis and Machine Intelligence, pages597-605, 1992, that an optimum set of generic, that is to sayproblem-independent, image operators for the continuous domain has thefollowing one-dimensional form: ##EQU1## where Φ_(n) (ξ) are theorthonormal eigenfunctions of the Schrodinger equations for thequantum-mechanical harmonic oscillator with the Hermite polynomialsH_(n) (ξ): ##EQU2## The scale-independent variable ξ is defined as##EQU3## and A(x,s) is a window function of the range of the scale swhich masks the section to be modelled: ##EQU4##

The image operator Ψ_(n) (x,s) can be interpreted such that a localimage section, which is defined by A(x,s) and weighted, is expanded inaccordance with a scale-independent orthonormal basis of the orthonormaleigenfunctions Φ_(n) (ξ) which define the local image structure.

On the other hand, the image operator Ψ_(n) (x,s) is identical, but fora constant factor, with the nth derivative of the operator of zerothorder. This permits the application of the image operator Ψ_(n) (x,s) tobe referred to as a Gaussian-weighted derivative operator.

The Model in the Discrete Domain

The image processing is typically performed in a discrete raster and notin the continuum. In order to put the concept into practice, it isnecessary to formulate the theory for the one-dimensional continuousdomain at least approximately in the discrete domain, without losing thedecisive properties. This has not been possible to date.

In particular, the aim is to fulfil the condition of orthonormality inthe discrete domain. A few other details such as, for example, theconstant coefficients can, however, be sacrificed, in order to renderthe discrete theory consistent. The most obvious approximation,specifically to scan the continuous filter functions at discrete points,does not achieve the aim. Going down this path destroys theorthonormality, and filters such as, for example, the Laplace operator,whose impulse response should be free of the mean value, no longer havethis important property.

The key to the discretization is the approximation of the Gaussian coreby binomials, which are known to represent the best possible discreteapproximation to the Gaussian function. In particular, there arecorresponding scaling properties: whereas, in the case of folding with aGaussian core, a Gaussian core in turn produces a Gaussian core, in thecase of binomial coefficients the same recursion is produced as theresult of the propagation in the Pascal triangle: explicitly, forexample, the folding of two binomials of 2nd order produces121×121=14641. It holds in general that: B_(n) ×B_(m) =B_(n+m).Consequently, the discrete weighting function and the basis of zerothorder are completely determined by the requirement for an identicalvariance of the Gaussian and the binomial distributions, which ischaracterized by the following simple relationship:

    s=m/2

where s is a continuous scale parameter of the Gaussian core (see above)and m is a discrete order of the binomial.

The higher orders of the discrete versions are determined by making useof the circumstance that the image operators Ψ can be interpreted as aderivative but for a constant. In the discrete case, the derivatives aredetermined by least-square estimation of corresponding Taylorcoefficients in binomially weighted data. A method for simple andefficient determination of these discrete derivatives is described in M.Hashimoto and J. Sklansky, "Multiple-order derivatives for detectinglocal image characteristics", Computer Vision, Graphics and ImageProcessing, 39: pages 28-55, 1987.

These measures guarantee the orthogonality of the fundamental discretebasis. Orthonormality is achieved by the explicit requirement ofnumerical standardization.

In order to achieve a conclusive and simple formulation, the continuousformulae are further used below. However, the discrete transformationsdescribed above must be used in the application.

2D-generalization

The two-dimensional (2D) generalization is also described in theabove-named publication of Generic neighbourhood operators, 1992. Thereare three different representations in two dimensions, of which each hasits own special properties. All these representations can be transformedinto one another by simple unitary algebraic transformations. This canbe achieved in a simple way by considering the fact that theeigenfunctions of the harmonic oscillator define a Hilbert space inwhich each orthonormal coordinate system can be transformed into anotherby rotation.

1. The Cartesian separable representation is the most interesting withregard to calculability, because the application of the 2D-operator canbe decomposed into the application of two one-dimensional (1D)operators:

    C.sub.pq (x,y,s)=Ψ.sub.p (x,s)*Ψ.sub.q (y,s)

The different operators are arranged in families of the same ordern=p+q. These are shown in FIG. 3.

In the case of consideration of a discrete filter mask having aneffective size of 20 pixels, the decomposition produces a rise in theefficiency by a factor of 10 (20+20 instead of 20*20 elementaryoperations).

2. The polar-separable representation is the representation best suitedfor scaling transformations or rotation transformations, because thescale affects only the radial component, whereas changes in theorientation affect only the angular component.

3. The so-called wavetrain representation is the best suited forapplication in the case of fingerprint analysis. The members of a familyare only rotated versions of the smoothed 1D-directional derivative. Bycontrast with the two other representations, the individual members arenot weighted, orthonormal basic functions. The decisive property isthat, for a given order n, the superposition of a set of n+1 prescribedfilter operations suffices to generate a directional filter whichfilters out arbitrary directions. In "Steerable filters for imageanalysis", W. T. Freeman and E. H. Adelson, Technical Report 126, MITMedia Lab., 1990, this phenomenon is referred to as steerability of suchfilters. The steerability can be demonstrated by an addition theorem ofHermite polynomials. The wavetrain representation greatly resembles

a) Gabor filters, which are used in the prior art of image processing,and

b) the response function of the direction-sensitive cells in the brainof higher mammals.

The associated wavetrain functions are shown in FIG. 4. Thedirection-specific character of the wavetrain functions is to be seen inthe figure.

The translation of the continuous 2D-theory into the discrete spacerequires a few considerations. Whereas the scaling response of theresulting basic functions follows directly from the corresponding(continuous) Gaussian bell and (discrete) binomials, the controllabilitycan only be approximated. With the realistic assumption that, given anadequately fine rastering, the deviations between the continuousfunction and the discrete approximation become negligible, the rotationcoefficients, which are used in the continuous domain, are alsooptimally suitable in the discrete domain. It is clear that it shouldnot be forgotten that the approximation of an arbitrarily rotatingdiscrete basic function is worsened when a relatively low binomial orderis present, because the number of the raster points is greater preciselyby 1 than the binomial order. For practical applications, the differencebetween the binomial order and the order of the maximally applicablederivative should be at least 3 to 4, in order to achieve an acceptablygood controllability and in order to exclude so-called "Aliasingeffects". This also guarantees the required band limitation, which isrequired in order to prevent modelling noise.

The wavetrains of fourth or fifth order are very well suited as localmodels of the linear/valley structure for the purpose of use infingerprint analysis, since they also take account of the twoneighbouring lines in the modelling.

Determination of the Optimum Orientation and the Magnitude Scale

Starting from the foundation of a wavetrain base of specific order on aspecific magnitude scale, it is to be expected that the absoluteresponse will be more intense when the wavetrain models are in optimumagreement with the linear/valley structure of a fingerprint. Whenmeasuring the actual angular response of a continuously rotatedwavetrain basic function of, for example, fourth degree, which isapplied in the case of a relatively good linear/valley structure, theresponse approximately obeys a cosine or sine function. Consequently,the squares of the wavetrain basic response functions are projected ontothis cosine and sine base with a period of 180°, in order to determinethe optimum local alignment, which leads to a directional field whosearc specifies the local orientation. On the basis of the fact that thelocal alignment changes mostly continuously (and gradually) in theimages, the resulting directional field can and should be stronglysmoothed.

The determination of the optimum magnitude scale is possible in asimilarly simple way. It requires filtering the image with a specificset of filters on all possibly relevant magnitude scales. The optimummagnitude scale is yielded as that whose filter delivers the signal withthe highest absolute value. The result is suitably interpolated when fewfilters are used in the case of fixed scales.

The Quality Measurement

The quality measurement can be best explained using the Cartesian basicfunctions in the 2D-domain. The multiplication properties which wererepresented above for the image operator Ψ are valid for the basicfunction Φ. It is sensible to arrange the 2D-functions in families ofdegree n, each having n+1 members, which is indicated below with theindex k.

A local partial image is expanded on the basis of the orthonormalcharacter of the basic functions Φ_(k),n+k, in accordance with the base:##EQU5##

The coefficients γ_(k),n-k are given by ##EQU6##

It holds, according to the Parseval theorem, that: ##EQU7##

In the case of discrete coordinates x and y, the infintely dimensionedHilbert space becomes a finitely dimensioned linear vector space. Thus,there are also only finitely many coefficients γ_(k),n-k. Their maximumnumber is the number of the data points occurring inside the windowdefined by A(x,y,s).

The coefficient γ₀,0²,, which represents the mean value, has a specialrole: In the Parseval theorem, it is frequently subtracted on both sidesof the equation, which results in the variance being equal to the sum ofthe squared coefficients, apart from the 0th. If only a subset S ofcoefficients is used as approximate value, its sum in relation to thevariance yields a value which is defined pointwise and is used as anobjective measure Q of quality in the sense of the variance defined bythe model. ##EQU8##

In the fingerprint model discussed above, only a single coefficient offourth degree is used as a local image model with the optimum magnitudescale and the optimum direction. The squared value of this singlecoefficient is used as the local measure of quality. This measure ofquality is normally much smaller than the maximum value of 1, because,even in the ideal case, in which parallel lines are present, the modelstructure decreases with increasing distance from the centre, while theideal image is periodic. A measure of quality of 0.3 is enough to beregarded as excellent. On the other hand, none of the coefficientshaving the above-described properties will overshoot the measure ofquality of 0.15 in response to a completely random signal. Consequently,a threshold value of 0.16 to 0.17 is a good lower bound for thehypothesis that a useful signal is present.

The typical local features of a fingerprint, the line ends andbranchings, yield local minima of the measure of quality, because thereare local errors in the model. Consequently, in order to describe thequality of these minutiae (MI), it is necessary to use an average valuefrom their neighbourhood.

A similar problem occurs in the case of the global singularities (SI) ofthe fingerprint. By definition, no preferred direction which can bespecified is present at these points. Consequently, the measure ofquality of a rigid 1D-model will necessarily vanish at these points.However, quality is not a pointwise property, but a regional one. Forthis reason, the measure of quality is determined for an environment ofsuitable size. This also has the effect that, at locations at whichspecial fingerprint features are present, there is no reduction in themeasure of quality, although they disturb the ideal model.

The total algorithm will now be described more precisely with the aid ofFIGS. 2 and 5 to 15. The concepts described are the core for thesolution of the problems which attend the coding of fingerprints, thecode of which is to be used in an automatically constructed database.The latter contains a few steps which can be classified:

Preprocessing for a global grey-scale value and contrast correction.

Determination of the preferred direction in the close environment ofeach pixel of the image.

Segmentation, for example by marking regions relevant for theidentification.

Identification of singularities (SI) in the directional field.

Determination of global characteristics for the purpose ofclassification.

Determination and compression of minutiae (MI) from the filtered image.

These steps are discussed in detail below and explained in conjunctionwith the application of the model-based filtering discussed above.

Preprocessing

A fingerprint is digitized by a scanner in terms of grey-scale valueswhich yield a typical, original image of the fingerprint, as shown inFIG. 5. Such an original image has large differences in brightness andcontrast. These differences are compensated for the purpose of optimizedfurther processing. For this purpose, a regional mean value is firstlydetermined and subtracted from the original image. The resulting imagefree of the mean value still has all the contrast differences. Thelatter are determined in a similar way as in the case of the human eyeby means of a psuedologarithmic transformation, as is disclosed in "Thepsuedo-logarithmic transformation for robust displacement estimation",J. Dengler and M. Schmidt, R. E. Groβkopf, editor, Mustererkennung(patent recognition) 1990, Informatik Fachberichte 254, pages 275-281,Berlin-Heidelberg-New York-Tokyo, 1990, Springer. The result of the twopreprocessing steps is shown in FIG. 6. It supplies an optimum startingpoint for further analysis, because all the local contrasts arepreserved and the weak contrasts are substantially enhanced.

Identification of the Preferred Direction in Each Pixel

The determination of the local preferred direction, which corresponds tothe direction of the local linear/valley structure, is of greatsignificance from many points of view in fingerprint analysis.

Firstly, a pixel of the preprocessed grey-scale image, including thepixels surrounding the selected pixel, is selected as the image section.As a rule, the two-dimensional image section comprises a region of 20*20pixels. The selected image section is defined and weighted by means ofthe two-dimensional binomial function A(x,s).

As described above, this weighted image function is expanded inaccordance with a discrete approximation of the orthonormal Cartesianeigenfunctions Φ_(n) (ξ) of the quantum-mechanical harmonic oscillator.In this case, because of the reasons mentioned above, only the family of4th order is used. The remaining, less decisive coefficients aredispensed with for reasons of efficiency.

The algebraic rotation of the expansion coefficients is performed withthe aid of the rotation theorem for Hermite polynomials. The Cartesiancoefficients are thereby transformed into coefficients of the wavetrainbase WB, as represented in FIG. 4. The square of the absolute value ofthese coefficients represents the degree of fit of each of the models,shown in FIG. 4, of the wavetrain base WB to the image section. Thecoefficient with the highest absolute value specifies the best-fittingwavetrain basis WB. A vector field approximation of the preferreddirection is undertaken from this wavetrain base WB by means ofsmoothing and fitting the absolute values of the coefficients toperiodic functions. The angle of best orientation and the quality of fitare determined from this approximation. The arc of the resulting vectorcoefficient, comprising two components, specifies the direction of thelocal linear orientation in the image section under consideration, andits absolute value specifies the local measure of quality.

The vector coefficient is stored in a memory, and a next pixel, forexample neighbouring, is selected in order to determine the vectorcoefficients thereof. If the vector coefficients of all the pixels ofthe relevant image section are determined in this way, statisticalerrors in the orientations are greatly reduced by smoothing over a largeare (averaging). The entire procedure is repeated for a further twomagnitude scales. For each pixel, linear superposition is used to selectthe best-fitting wavetrain basic function, and thus the vectorcoefficient whose measure of quality with regard to direction and linearspacing is the greatest. The weighting ω of the individual contributionsis yielded from the respective qualities q_(i) : ##EQU9## A field withthe optimum scales can be stored in a memory MEM1 as a field.Information on the average linear spacing of the fingerprint can befurther utilized later with the aid of this scale field.

When a set of basic functions of fourth order on three differentmagnitude scales, which cover the possible range of spacings between thelines, is used, the projections of the operator responses onto thecosine and sine components produce the field of directional vectorswhich is coded in grey-scale values in FIG. 9. It is clear that this isa vector field having a range of 180°, which means that a rotation of180° has no effect on any of the components. The arc of this vectorfield specifies the actual direction, as it is represented by grey-scalecoding of the angle in FIG. 10. It is to be noted that white and blackrepresent the same angle of 0°. The directional field, which is storedin a memory, is the most important information source for the globalcharacterization of the fingerprint.

A quality map, which is shown as a grey-scale image in FIG. 7 and storedin a memory, is formed from the absolute value of the directionalvectors. It is possible with the aid of this quality map to obtainvarious findings on the image of the fingerprint which are helpful forclassification purposes.

An important task in the identification of a real fingerprint is theseparation of the actual fingerprint from its environment. The relevantimage region can be determined with the aid of threshold-value treatmentof the quality map. The maximum random quality of 0.15 is used asthreshold value, that is to say, if the absolute value of the quality isgreater than 0.15, there is present in the case of the pixel underconsideration a structure which points to the presence of a fingerprint.The result of this threshold-value treatment is a segmentation map, asshown in FIG. 8.

Since the quality map is smoothed, the region of good quality iscoherent, or has at least large coherent regions. It can be seen thatthe segmentation acts in a way similar to human intuition.

The directional image is masked by the segmentation result, in order toavoid further processing of interfering properties such as, for example,background noise.

Determination of Singularities (SI) From the Directional Field

There are two different types of noteworthy points in the directionalfield. These special points, also termed singularities (SI), do not havea well-defined direction. The singularities (SI) can be characterized bythe standardized integral of the angular variation in the environmentalfield, which can be interpreted as topological loading. The theorem onwinding numbers from the functional theory is applied, as described, forexample, in "Analysing Oriented Patterns", M. Kass, A. Witkin, ComputerVision, Graphics and Image Processing, pages 362 to 385, 1987. The mostimportant singularities (SI) are the core regions, with a winding number0.5 having a positive sign, and the delta regions, with a winding number0.5 having a negative sign. FIG. 11 shows the automatically determinedsingularities SI in the original image of the fingerprint.

In regions of undefined alignment, false singularities SI can bedetermined by mistake. These are eliminated by a statistical test foreach singularity SI. In order to arrive at a positive test result, theremust be an adequate amount of direction-specific information in a closeenvironment of a putative singularity SI. The nature of thesingularities SI, their coordinates and spacings from one another arestored in a memory MEM1. When specifying the spacing, use is made of thelength unit of "line number" (ridgecount) which specifies the number ofthe lines between the singularities SI.

One core category, the arcuate cores, does not actually correspond tothe singularities SI in the directional field. They are characterized bya type of elevation. Such singularities SI are treated by a specialprocedure. In this procedure, the directional field and the quality mapare used to set an arcuate core at the points at which a maximum ispresent in the line curvatures and there is good right/left symmetry.

Model-based Reconstruction

A central starting point in the reliable analysis of fingerprint imageinformation is the filtering process which is required in order toexclude false features and in order to find the correct ones. Theapproach to this is model-based; the image is actually reconstructed bylinear superposition of the best-fitting wavetrain basic function offourth order. The result is an idealized reconstruction of thelinear/valley structure, as shown in FIG. 12. Well reconstructed regionsexhibit strong contrast, while in the case of less well constructedregions weaker contrast is present.

Feature Extraction

The aim and purpose of fingerprint analysis is to achieve reliableextraction of minutiae MI. These unchanging line ends and branchings areof central significance in the identification of a person.

Starting from the above-described model-based reconstruction, the restof the method is trivial. Owing to the fact that all the operators offirst and higher order have an arithmetic mean of zero, the optimumthreshold value for the creation of a binary image is zero. For example,white is set for functional values of the reconstruction image which aregreater than zero, and black is set for functional values which are lessthan zero. The masked binary reconstruction in accordance with FIG. 13shows the excellent quality of the method represented.

Outlining

An applicable method for outlining is disclosed, for example, in"Morphologische Bildverarbeitung in der Zellenanalyse" ("Morphologicalimage processing in cell analysis"), M. Schmidt, Technical Report9-2/88, Deutsches Krebsforschungszentrum Heidelberg/DE (German CancerResearch Centre, Heidelberg), Medizinische und Biologische Informatik,1988. The aim of outlining is to generate lines of pixel width, as shownin FIG. 14. For this purpose, the line width is reduced in steps to thewidth of a pixel, without changing the topological context.

Line ends and branchings can be determined in a simple way from theoutlined binary image by a simple 3×3 pixel pattern comparison method(template matching). Short line segments resulting from the outliningmethod and having ends are subsequently removed. The fingerprintfeatures designated as minutiae MI are stored in the form of a list ofcoordinates, direction, type and quality.

As a result of the coding method, copious information on the fingerprintis available. This result is represented in FIG. 15, in which the globalsingularities SI and the minutiae MI are marked. The items ofinformation represented by the image are ordered and conditioned for thecomparator MT in a manner which is not represented, and stored in thememories MEM, MEM1.

The invention is not limited to the particular details of the apparatusdepicted and other modifications and applications are contemplated.Certain other changes may be made in the above described apparatuswithout departing from the true spirit and scope of the invention hereininvolved. It is intended, therefore, that the subject matter in theabove depiction shall be interpreted as illustrative and not in alimiting sense.

We claim:
 1. A method for reconstructing linear structures present inraster form, comprising the steps of:(a) selecting an image section,arranged around a pixel, from a linear-structure image which is presentin pixel form and digitized in terms of grey-scale values of gray-scaleinformation; (b) locally simulating the grey-scale information using aone-dimensional basic function Ψ_(n) (x,s), which is brought intorelation with a two-dimensional weighted binomial function A(x,s) thatdefines the image section, using: ##EQU10## (c) determining a preferreddirection, present for the selected pixel in conjunction with anenvironment thereof, of the lines running through the image point,by(c)(i) expanding the basic function Ψ_(n) (x,s), which is aGauss-weighted derivation operator, according to a discreteapproximation using binomials, according to the relation s=m/2, where sis a continuous scale parameter of a Gauss core and m is a discreteorder of the binomial, (c)(ii) selecting an expansion coefficient of anorder that enables a good approximation of the basic function to thegrey-scale information, (c)(iii) algebraically rotating the expansioncoefficients using a rotation theorem for Hermite polynomials, so thatcartesian coefficients are transformed into coefficients of a wave trainbase, (c)(iv) adapting coefficient absolute values to periodic functionof models of a wave train base, (d) storing the result, which contains avector coefficient with two components, as a directional distribution ina memory, and (e) selecting a next pixel and repeating steps (a) to (d)until the directional distribution of the pixels of the linear structureis stored in the memory.
 2. The method according to claim 1, wherein astatistical smoothing of the vector coefficients determined is carriedout using averaging.
 3. The method according to claim 1, wherein themethod is repeated using at least a second binomial function which isassigned to a different magnitude scale describing line spacings.
 4. Themethod according to claim 3, wherein with a selection of respectivelybest-fitting basic function, and thus of the vector coefficient, whosequality measure q with respect to direction and line spacing isgreatest, and weighting by production of a weighting function ##EQU11##of the individual absolute values, whereby the respective selectionresult is stored in a memory as a grey-scale image and as a directionaldistribution.
 5. The method according to claim 1, wherein the absolutevalue of each vector coefficient is formed and is stored as a qualitymap that visualizes the quality of a reconstruction of a linearstructure.
 6. The method according to claim 5, wherein, withsegmentation of a reconstruction result by threshold-value treatment ofthe quality map, all pixels having a quality which is higher than athreshold value are assigned to a valid linear structure.
 7. The methodaccording to claim 1, wherein, with determination of singularities froma stored directional distribution with formation of a normed integral ofangular change of an environmental field.
 8. The method according toclaim 7, wherein the method further comprises checking of existingvector coefficients in an environment of a singularity and storing datafor finding a singularity recognized as authentic.
 9. The methodaccording to claim 1, wherein the method further comprisesreconstructing all pixels from a grey-scale image to form a binary imageby setting a limiting value for distinguishing between white and blackat a grey-scale value of
 0. 10. The method according to claim 8, whereinthe method further comprises outlining of a binary image so that a linewidth is reduced in steps to a width of a pixel.
 11. The methodaccording to claim 10, wherein the method further comprises:sequentiallyselecting pixel image sections, of size 3×3 pixels each, from theoutlined binary image, comparing said pixel image sections with pixelpatterns of a common size which contain branchings or line ends, andstoring image point coordinates for a positive result of comparison. 12.The method according to claim 11, wherein the method further comprisesevaluating the reliability of a found minutia using quality measures,stored in the quality map, in an environment of each minutia.
 13. Adevice for reconstructing linear structures present in raster formcomprising:system for selecting an image section, arranged around apixel, from a linear-structure image which is present in pixel form anddigitized in terms of grey-scale values; system for local simulation ofthe grey-scale information using a one-dimensional basic function Ψ_(n)(x,s) that defines the image section, which is brought into relationwith a two-dimensional weighted binomial function A(x,s) using:##EQU12## system for determining a preferred direction, present for theselected pixel in conjunction with an environment thereof, of the linesrunning through the image point, by, expanding the basic function Ψ_(n)(x,s), which is a Gauss-weighted derivation operator, according to adiscrete approximation using binomials, according to the relation s=m/2,where s is a continuous scale parameter of a Gauss core and m is adiscrete order of the binomial, selecting an expansion coefficient of anorder that enables a good approximation of the basic function to thegrey-scale value information, algebraically rotating the expansioncoefficients using rotation theorem for Hermite polynomials, so thatcertesian coefficients are converted into coefficients of a wave trainbase WB, adapting the coefficient absolute values to the periodicfunction of models of a wave train base, system for storing the result,which contains a vector coefficient with two components, as adirectional distribution in a memory, and system for repeatedlyselecting and processing a next pixel until the directional distributionof the pixels of the linear structure is stored in the memory.
 14. Thedevice according to claim 13, wherein the device further comprises,smoothing the vector coefficients determined using averaging.
 15. Thedevice according to claim 13, wherein the device further comprises asystem for repetition of method steps by at least a second binomialfunction which is assigned to a different magnitude scale describingline spacings.
 16. The device according to claim 15, wherein the devicefurther comprises a system for selecting a respectively best-fittingbasic function, and thus of the vector coefficient, whose qualitymeasure q with respect to direction and line spacing is greatest, andweighting by production of a weighting function ##EQU13## of theindividual absolute values, whereby a respective selection result isstored in a memory as a grey-scale image and as a directionaldistribution.
 17. The device according to claim 13, wherein the devicefurther comprises system elements for forming the absolute value of eachvector coefficient and for storage in a form of a quality map thatvisualizes a quality of reconstruction of a linear structure.
 18. Thedevice according to claim 17, wherein the device further comprises asystem for segmenting the reconstruction result by threshold-valuetreatment of the quality map, wherein all pixels having a quality whichis higher than a threshold value are assigned to a valid linearstructure.
 19. The device according to claim 13, wherein the devicefurther comprises a system for determining singularities from the storeddirectional distribution, with formation of normed integral of angularchange of an environmental field.
 20. The device according to claim 19,wherein the device further comprises a system for checking existingvector coefficients in an environment of a singularity recognized asauthentic.
 21. The device according to claim 13 wherein the devicefurther comprises a system for reconstructing all pixels from agrey-scale image to form a binary image by setting the limiting valuefor distinguishing between white and black at a grey-scale value of 0.22. The device according to claim 20, wherein the device furthercomprises a system for outlining of a binary image so that a line widthis reduced in steps of a width of a pixel.
 23. The device according toclaim 22, wherein the device further comprises:a system for sequentialselection of pixel image sections, of size 3×3 pixels each, from theoutlined binary image, comparison of these pixel image sections withpixel patterns of a common size which contain branchings or line ends,and storage of image point coordinates for a positive result ofcomparison.
 24. The device according to claim 23, wherein the devicefurther comprises a system for evaluating releabilty of a found minutiausing quality measures, stored in the quality map, in an environment pfeach minutia.
 25. The method according to claim 1, wherein the linearstructures are fingerprints.
 26. The device according to claim 13,wherein the linear structures are fingerprints.